Let $\mathbb{Q}(\sqrt{2})$ be the set of real numbers of the form $a + b\sqrt{2}$ where $a, b \in \mathbb{Q}$. Let $+$ denote standard addition and $*$ denote standard multiplication. We will show that then $\mathbb{Q}(\sqrt{2})$ forms a ring with these operations by verifying each of the axioms above.